The numerical schemes which will be presented below include most of the mentioned works on local and nonlocal cases. However, none of the above showed convergence in \(C([0,T];L^p(\mathbb {R}^N))\)  (but some could still deduce that the limit itself belonged to that space, see also Section REF ). Our equitightness estimates therefore improves convergence results already present in the literature. Conservative and monotone finite-difference schemes for scalar conservation laws are discussed in [1]}, and they immediately fall into our \(C([0,T];L^p(\mathbb {R}^N))\) -framework. By adding a possibly nonlinear local diffusion term to such equations, we obtain convection-diffusion equations with local diffusion. Such equations have been studied in the context of finite-difference approximations in [2]}. See also Theorem 3.9 and Corollary 3.10 in [3]}. Again, we can improve the convergence from the respective \(L_\textup {loc}^1(Q_T)\)  and \(C([0,T];L_\textup {loc}^1(\mathbb {R}^N))\)  to \(C([0,T];L^1(\mathbb {R}^N))\)  since \(L^1\) -equitightness holds. In the nonlocal diffusion setting, finite-difference schemes has just recently been analyzed in rigorous detail and generality in [4]}, [5]}, see also [6]}, [7]}. The former two references obtain compactness/convergence results in \(C([0,T];L_\textup {loc}^1(\mathbb {R}^N))\) , and the latter in \(L_\textup {loc}^1(Q_T)\) . Our framework thus improves the compactness/convergence results of all four papers.
